Theorem recexprlemm 7586

Description: B is inhabited. Lemma for recexpr 7600. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.

Assertion

Ref	Expression
recexprlemm	|- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )

Distinct variable groups:   r, q, x, y, A   B, q, r, x, y

Proof of Theorem recexprlemm

Step	Hyp	Ref	Expression
1		prop 7437	. . 3 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7440	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 2nd ` A ) )
3		recclnq 7354	. . . . . . 7 |- ( x e. Q. -> ( *Q ` x ) e. Q. )
4		nsmallnqq 7374	. . . . . . 7 |- ( ( *Q ` x ) e. Q. -> E. q e. Q. q <Q ( *Q ` x ) )
5	3, 4	syl 14	. . . . . 6 |- ( x e. Q. -> E. q e. Q. q <Q ( *Q ` x ) )
6	5	adantr 274	. . . . 5 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> E. q e. Q. q <Q ( *Q ` x ) )
7		recrecnq 7356	. . . . . . . . . . . 12 |- ( x e. Q. -> ( *Q ` ( *Q ` x ) ) = x )
8	7	eleq1d 2239	. . . . . . . . . . 11 |- ( x e. Q. -> ( ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) <-> x e. ( 2nd ` A ) ) )
9	8	anbi2d 461	. . . . . . . . . 10 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) <-> ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) ) )
10		breq2 3993	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( q <Q y <-> q <Q ( *Q ` x ) ) )
11		fveq2 5496	. . . . . . . . . . . . . 14 |- ( y = ( *Q ` x ) -> ( *Q ` y ) = ( *Q ` ( *Q ` x ) ) )
12	11	eleq1d 2239	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( ( *Q ` y ) e. ( 2nd ` A ) <-> ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) )
13	10, 12	anbi12d 470	. . . . . . . . . . . 12 |- ( y = ( *Q ` x ) -> ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) ) )
14	13	spcegv 2818	. . . . . . . . . . 11 |- ( ( *Q ` x ) e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
15	3, 14	syl 14	. . . . . . . . . 10 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
16	9, 15	sylbird 169	. . . . . . . . 9 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
17		recexpr.1	. . . . . . . . . 10 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
18	17	recexprlemell 7584	. . . . . . . . 9 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
19	16, 18	syl6ibr 161	. . . . . . . 8 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) -> q e. ( 1st ` B ) ) )
20	19	expcomd 1434	. . . . . . 7 |- ( x e. Q. -> ( x e. ( 2nd ` A ) -> ( q <Q ( *Q ` x ) -> q e. ( 1st ` B ) ) ) )
21	20	imp 123	. . . . . 6 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> ( q <Q ( *Q ` x ) -> q e. ( 1st ` B ) ) )
22	21	reximdv 2571	. . . . 5 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> ( E. q e. Q. q <Q ( *Q ` x ) -> E. q e. Q. q e. ( 1st ` B ) ) )
23	6, 22	mpd 13	. . . 4 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> E. q e. Q. q e. ( 1st ` B ) )
24	23	rexlimiva 2582	. . 3 |- ( E. x e. Q. x e. ( 2nd ` A ) -> E. q e. Q. q e. ( 1st ` B ) )
25	1, 2, 24	3syl 17	. 2 |- ( A e. P. -> E. q e. Q. q e. ( 1st ` B ) )
26		prml 7439	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 1st ` A ) )
27		1nq 7328	. . . . . . . 8 |- 1Q e. Q.
28		addclnq 7337	. . . . . . . 8 |- ( ( ( *Q ` x ) e. Q. /\ 1Q e. Q. ) -> ( ( *Q ` x ) +Q 1Q ) e. Q. )
29	3, 27, 28	sylancl 411	. . . . . . 7 |- ( x e. Q. -> ( ( *Q ` x ) +Q 1Q ) e. Q. )
30		ltaddnq 7369	. . . . . . . 8 |- ( ( ( *Q ` x ) e. Q. /\ 1Q e. Q. ) -> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) )
31	3, 27, 30	sylancl 411	. . . . . . 7 |- ( x e. Q. -> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) )
32		breq2 3993	. . . . . . . 8 |- ( r = ( ( *Q ` x ) +Q 1Q ) -> ( ( *Q ` x ) <Q r <-> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) ) )
33	32	rspcev 2834	. . . . . . 7 |- ( ( ( ( *Q ` x ) +Q 1Q ) e. Q. /\ ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) ) -> E. r e. Q. ( *Q ` x ) <Q r )
34	29, 31, 33	syl2anc 409	. . . . . 6 |- ( x e. Q. -> E. r e. Q. ( *Q ` x ) <Q r )
35	34	adantr 274	. . . . 5 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> E. r e. Q. ( *Q ` x ) <Q r )
36	7	eleq1d 2239	. . . . . . . . . . 11 |- ( x e. Q. -> ( ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) <-> x e. ( 1st ` A ) ) )
37	36	anbi2d 461	. . . . . . . . . 10 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) <-> ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) ) )
38		breq1 3992	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( y <Q r <-> ( *Q ` x ) <Q r ) )
39	11	eleq1d 2239	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) )
40	38, 39	anbi12d 470	. . . . . . . . . . . 12 |- ( y = ( *Q ` x ) -> ( ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) ) )
41	40	spcegv 2818	. . . . . . . . . . 11 |- ( ( *Q ` x ) e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
42	3, 41	syl 14	. . . . . . . . . 10 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
43	37, 42	sylbird 169	. . . . . . . . 9 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
44	17	recexprlemelu 7585	. . . . . . . . 9 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
45	43, 44	syl6ibr 161	. . . . . . . 8 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) -> r e. ( 2nd ` B ) ) )
46	45	expcomd 1434	. . . . . . 7 |- ( x e. Q. -> ( x e. ( 1st ` A ) -> ( ( *Q ` x ) <Q r -> r e. ( 2nd ` B ) ) ) )
47	46	imp 123	. . . . . 6 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> ( ( *Q ` x ) <Q r -> r e. ( 2nd ` B ) ) )
48	47	reximdv 2571	. . . . 5 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> ( E. r e. Q. ( *Q ` x ) <Q r -> E. r e. Q. r e. ( 2nd ` B ) ) )
49	35, 48	mpd 13	. . . 4 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> E. r e. Q. r e. ( 2nd ` B ) )
50	49	rexlimiva 2582	. . 3 |- ( E. x e. Q. x e. ( 1st ` A ) -> E. r e. Q. r e. ( 2nd ` B ) )
51	1, 26, 50	3syl 17	. 2 |- ( A e. P. -> E. r e. Q. r e. ( 2nd ` B ) )
52	25, 51	jca 304	1 |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156   E.wrex 2449   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   1Qc1q 7243   +Q cplq 7244   *Qcrq 7246   <Q cltq 7247   P.cnp 7253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428

This theorem is referenced by:  recexprlempr  7594